Problem: Augustus draws tickets one at a time for a raffle. The person named on the ticket must be present to win, but $30\%$ of the $750$ raffle tickets have the names of people who are no longer present. Let $T$ be the number of tickets Augustus needs to draw to find a winner who is present. Find the probability that Augustus first draws the name of someone present on the $3^{\text{rd}}$ ticket. You may round your answer to the nearest hundredth. $P(T=3)=$
Solution: Without a fancy calculator For each ticket: $P({\text{present}})=0.7$ $P(\text{gone}})=0.3$ If Augustus first draws the name of someone present on the $3^{\text{rd}}$ ticket, his sequence of names on the tickets must be "gone, gone, present." Since we are sampling less than $10\%$ of the population of tickets, we can assume independence and multiply the probabilities. $\begin{aligned} P(T=3)&=P(\text{gone}}, \text{gone}}, {\text{present}}) \\\\ &=(0.3})(0.3})({0.7}) \\\\ &=(0.3)^2(0.7) \\\\ &=0.063 \end{aligned}$ $P(T=3) =0.063$